E is irrational, Mathematics

Assignment Help:

If e were rational, then e = n/m for some positive integers m, n. So then 1/e = m/n. But the series expansion for 1/e is

1/e = 1 - 1/1! + 1/2! - 1/3! + ...

Call the first n values of this alternating series S(n). How good is this approximation to e? Well, the error is bounded by the next part of the alternating series:

0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n+1)!

But multiplying through by n!, you can see that

0 < | integer - integer | < 1/(n+1) < 1.

But there is no integer between 0 and 1, so this is a contradiction; e must be irrational.


Related Discussions:- E is irrational

Find the derivatives, Find the derivatives of the following functions a)...

Find the derivatives of the following functions a)      y = 5x 4 +3x -1-x 3 b)      y = (x+1) -1/2 c)      y= e x2+1 d)     y= e 3x lnx e)     y =ln(x+1/x)y

Fourier series - partial differential equations, Fourier series - Partial D...

Fourier series - Partial Differential Equations One more application of series arises in the study of Partial Differential Equations.  One of the more generally employed method

Expertes, how to do multiplication

how to do multiplication

Dr.., I need some material on Bachet equation

I need some material on Bachet equation

Jay bought twenty-five $0.37 stamps how much did he spend, Jay bought twent...

Jay bought twenty-five $0.37 stamps. How much did he spend? To ?nd how much Jay spent, you must multiply the cost of each stamp ($0.37) through the number of stamps purchased (

Compare and contrast african immigrants, Compare and contrast African immig...

Compare and contrast African immigrants with our immigrant groups? How are they different? What are the implications of these differences for their adjustment to the larger society

Create a circular table with no restrictions, 1. Four different written dri...

1. Four different written driving tests are administered by a city. One of these tests is selected at random for each applicant for a drivers license. If a group of 2 women and 4 m

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd